↑

Properties

Label 512.2.k.a.497.10
Level $512$
Weight $2$
Character 512.497
Analytic conductor $4.088$
Analytic rank $0$
Dimension $240$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [512,2,Mod(17,512)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("512.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(512, base_ring=CyclotomicField(32)) chi = DirichletCharacter(H, H._module([0, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 512 = 2^{9} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 512.k (of order \(32\), degree \(16\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.08834058349\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(15\) over \(\Q(\zeta_{32})\)
Twist minimal: no (minimal twist has level 128)
Sato-Tate group: $\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label 497.10
Character \(\chi\) \(=\) 512.497
Dual form 512.2.k.a.273.10

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.12781 - 0.602827i) q^{3} +(-1.29871 - 1.58248i) q^{5} +(1.93875 - 1.29543i) q^{7} +(-0.758155 + 1.13466i) q^{9} +(4.58482 - 1.39079i) q^{11} +(-1.77365 - 1.45559i) q^{13} +(-2.41866 - 1.00184i) q^{15} +(-0.698095 + 0.289160i) q^{17} +(-0.355280 - 3.60722i) q^{19} +(1.40562 - 2.62973i) q^{21} +(0.824744 - 0.164052i) q^{23} +(0.157851 - 0.793571i) q^{25} +(-0.547088 + 5.55468i) q^{27} +(2.64076 - 8.70543i) q^{29} +(-4.31573 - 4.31573i) q^{31} +(4.33241 - 4.33241i) q^{33} +(-4.56786 - 1.38565i) q^{35} +(-3.24347 - 0.319454i) q^{37} +(-2.87781 - 0.572431i) q^{39} +(2.34453 + 11.7868i) q^{41} +(7.17325 + 3.83418i) q^{43} +(2.78020 - 0.273825i) q^{45} +(1.13781 + 2.74691i) q^{47} +(-0.598174 + 1.44412i) q^{49} +(-0.613005 + 0.746949i) q^{51} +(2.85064 + 9.39731i) q^{53} +(-8.15524 - 5.44916i) q^{55} +(-2.57522 - 3.85409i) q^{57} +(3.31394 - 2.71968i) q^{59} +(0.0672870 + 0.125885i) q^{61} +3.18196i q^{63} +4.69715i q^{65} +(-3.12347 - 5.84360i) q^{67} +(0.831260 - 0.682197i) q^{69} +(6.30859 + 9.44148i) q^{71} +(-4.87067 - 3.25448i) q^{73} +(-0.300360 - 0.990154i) q^{75} +(7.08715 - 8.63572i) q^{77} +(-4.87747 + 11.7752i) q^{79} +(1.16482 + 2.81213i) q^{81} +(13.3221 - 1.31211i) q^{83} +(1.36421 + 0.729186i) q^{85} +(-2.26959 - 11.4100i) q^{87} +(-13.5456 - 2.69438i) q^{89} +(-5.32428 - 0.524395i) q^{91} +(-7.46896 - 2.26568i) q^{93} +(-5.24694 + 5.24694i) q^{95} +(4.07614 + 4.07614i) q^{97} +(-1.89793 + 6.25664i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 240 q + 16 q^{3} - 16 q^{5} + 16 q^{7} - 16 q^{9} + 16 q^{11} - 16 q^{13} + 16 q^{15} - 16 q^{17} + 16 q^{19} - 16 q^{21} + 16 q^{23} - 16 q^{25} + 16 q^{27} - 16 q^{29} + 16 q^{31} - 16 q^{33} + 16 q^{35}+ \cdots + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/512\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(511\)
\(\chi(n)\) \(e\left(\frac{9}{32}\right)\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.12781 − 0.602827i 0.651142 − 0.348043i −0.112529 − 0.993648i \(-0.535895\pi\)
0.763671 + 0.645606i \(0.223395\pi\)
\(4\) 0 0
\(5\) −1.29871 − 1.58248i −0.580800 − 0.707706i 0.396865 − 0.917877i \(-0.370098\pi\)
−0.977665 + 0.210171i \(0.932598\pi\)
\(6\) 0 0
\(7\) 1.93875 − 1.29543i 0.732779 − 0.489627i −0.132334 − 0.991205i \(-0.542247\pi\)
0.865112 + 0.501578i \(0.167247\pi\)
\(8\) 0 0
\(9\) −0.758155 + 1.13466i −0.252718 + 0.378220i
\(10\) 0 0
\(11\) 4.58482 − 1.39079i 1.38238 − 0.419339i 0.490430 − 0.871481i \(-0.336840\pi\)
0.891946 + 0.452141i \(0.149340\pi\)
\(12\) 0 0
\(13\) −1.77365 − 1.45559i −0.491921 − 0.403709i 0.355463 − 0.934690i \(-0.384323\pi\)
−0.847383 + 0.530982i \(0.821823\pi\)
\(14\) 0 0
\(15\) −2.41866 − 1.00184i −0.624495 − 0.258674i
\(16\) 0 0
\(17\) −0.698095 + 0.289160i −0.169313 + 0.0701317i −0.465730 − 0.884927i \(-0.654208\pi\)
0.296417 + 0.955059i \(0.404208\pi\)
\(18\) 0 0
\(19\) −0.355280 − 3.60722i −0.0815068 − 0.827553i −0.945840 − 0.324633i \(-0.894759\pi\)
0.864333 − 0.502920i \(-0.167741\pi\)
\(20\) 0 0
\(21\) 1.40562 − 2.62973i 0.306732 − 0.573855i
\(22\) 0 0
\(23\) 0.824744 − 0.164052i 0.171971 − 0.0342071i −0.108354 − 0.994112i \(-0.534558\pi\)
0.280325 + 0.959905i \(0.409558\pi\)
\(24\) 0 0
\(25\) 0.157851 − 0.793571i 0.0315702 − 0.158714i
\(26\) 0 0
\(27\) −0.547088 + 5.55468i −0.105287 + 1.06900i
\(28\) 0 0
\(29\) 2.64076 − 8.70543i 0.490378 − 1.61656i −0.267080 − 0.963674i \(-0.586059\pi\)
0.757458 − 0.652884i \(-0.226441\pi\)
\(30\) 0 0
\(31\) −4.31573 − 4.31573i −0.775127 − 0.775127i 0.203871 − 0.978998i \(-0.434648\pi\)
−0.978998 + 0.203871i \(0.934648\pi\)
\(32\) 0 0
\(33\) 4.33241 − 4.33241i 0.754175 − 0.754175i
\(34\) 0 0
\(35\) −4.56786 − 1.38565i −0.772110 − 0.234217i
\(36\) 0 0
\(37\) −3.24347 − 0.319454i −0.533224 − 0.0525179i −0.172178 − 0.985066i \(-0.555080\pi\)
−0.361046 + 0.932548i \(0.617580\pi\)
\(38\) 0 0
\(39\) −2.87781 − 0.572431i −0.460818 − 0.0916624i
\(40\) 0 0
\(41\) 2.34453 + 11.7868i 0.366154 + 1.84078i 0.521929 + 0.852989i \(0.325213\pi\)
−0.155775 + 0.987793i \(0.549787\pi\)
\(42\) 0 0
\(43\) 7.17325 + 3.83418i 1.09391 + 0.584707i 0.916745 − 0.399474i \(-0.130807\pi\)
0.177166 + 0.984181i \(0.443307\pi\)
\(44\) 0 0
\(45\) 2.78020 − 0.273825i 0.414447 − 0.0408195i
\(46\) 0 0
\(47\) 1.13781 + 2.74691i 0.165966 + 0.400678i 0.984880 − 0.173239i \(-0.0554232\pi\)
−0.818913 + 0.573917i \(0.805423\pi\)
\(48\) 0 0
\(49\) −0.598174 + 1.44412i −0.0854535 + 0.206303i
\(50\) 0 0
\(51\) −0.613005 + 0.746949i −0.0858379 + 0.104594i
\(52\) 0 0
\(53\) 2.85064 + 9.39731i 0.391566 + 1.29082i 0.902310 + 0.431087i \(0.141870\pi\)
−0.510745 + 0.859732i \(0.670630\pi\)
\(54\) 0 0
\(55\) −8.15524 − 5.44916i −1.09965 − 0.734764i
\(56\) 0 0
\(57\) −2.57522 − 3.85409i −0.341096 − 0.510486i
\(58\) 0 0
\(59\) 3.31394 − 2.71968i 0.431439 − 0.354073i −0.393471 − 0.919337i \(-0.628726\pi\)
0.824910 + 0.565264i \(0.191226\pi\)
\(60\) 0 0
\(61\) 0.0672870 + 0.125885i 0.00861521 + 0.0161179i 0.886192 − 0.463318i \(-0.153341\pi\)
−0.877577 + 0.479436i \(0.840841\pi\)
\(62\) 0 0
\(63\) 3.18196i 0.400889i
\(64\) 0 0
\(65\) 4.69715i 0.582609i
\(66\) 0 0
\(67\) −3.12347 − 5.84360i −0.381593 − 0.713910i 0.615658 − 0.788014i \(-0.288890\pi\)
−0.997251 + 0.0741040i \(0.976390\pi\)
\(68\) 0 0
\(69\) 0.831260 − 0.682197i 0.100072 − 0.0821269i
\(70\) 0 0
\(71\) 6.30859 + 9.44148i 0.748692 + 1.12050i 0.988727 + 0.149733i \(0.0478413\pi\)
−0.240034 + 0.970764i \(0.577159\pi\)
\(72\) 0 0
\(73\) −4.87067 − 3.25448i −0.570069 − 0.380908i 0.236871 − 0.971541i \(-0.423878\pi\)
−0.806940 + 0.590633i \(0.798878\pi\)
\(74\) 0 0
\(75\) −0.300360 − 0.990154i −0.0346826 − 0.114333i
\(76\) 0 0
\(77\) 7.08715 − 8.63572i 0.807656 − 0.984132i
\(78\) 0 0
\(79\) −4.87747 + 11.7752i −0.548758 + 1.32482i 0.369646 + 0.929173i \(0.379479\pi\)
−0.918403 + 0.395645i \(0.870521\pi\)
\(80\) 0 0
\(81\) 1.16482 + 2.81213i 0.129425 + 0.312459i
\(82\) 0 0
\(83\) 13.3221 − 1.31211i 1.46229 − 0.144023i 0.664649 − 0.747155i \(-0.268581\pi\)
0.797641 + 0.603132i \(0.206081\pi\)
\(84\) 0 0
\(85\) 1.36421 + 0.729186i 0.147970 + 0.0790914i
\(86\) 0 0
\(87\) −2.26959 − 11.4100i −0.243326 − 1.22328i
\(88\) 0 0
\(89\) −13.5456 − 2.69438i −1.43583 − 0.285604i −0.584993 − 0.811038i \(-0.698903\pi\)
−0.850834 + 0.525435i \(0.823903\pi\)
\(90\) 0 0
\(91\) −5.32428 − 0.524395i −0.558136 − 0.0549716i
\(92\) 0 0
\(93\) −7.46896 − 2.26568i −0.774495 − 0.234940i
\(94\) 0 0
\(95\) −5.24694 + 5.24694i −0.538325 + 0.538325i
\(96\) 0 0
\(97\) 4.07614 + 4.07614i 0.413870 + 0.413870i 0.883084 − 0.469214i \(-0.155463\pi\)
−0.469214 + 0.883084i \(0.655463\pi\)
\(98\) 0 0
\(99\) −1.89793 + 6.25664i −0.190749 + 0.628816i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 512.2.k.a.497.10 240
4.3 odd 2 128.2.k.a.101.3 ✓ 240
128.19 odd 32 128.2.k.a.109.3 yes 240
128.109 even 32 inner 512.2.k.a.273.10 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.101.3 ✓ 240 4.3 odd 2
128.2.k.a.109.3 yes 240 128.19 odd 32
512.2.k.a.273.10 240 128.109 even 32 inner
512.2.k.a.497.10 240 1.1 even 1 trivial